Abstract
We study the moments of the k-th nearest neighbor distance for independent identically distributed points in $\mathbb{R}^n$. In the earlier literature, the case with power higher than n has been analyzed by assuming a bounded support for the underlying density. The boundedness assumption is removed by assuming the multivariate Gaussian distribution. In this case, the nearest neighbor distances show very different behavior in comparison to earlier results.
Highlights
Consider a set of independent identically distributed (i.i.d.) random variables (Xi)M i=1 with a common density p(x) on Rn
We study the moments of the nearest neighbor distance
The quantity (1) appears commonly in the literature on random geometric graphs, where directed and undirected nearest neighbor graphs are analyzed as special cases of more general frameworks [9, 10, 13]
Summary
Consider a set of independent identically distributed (i.i.d.) random variables (Xi)M i=1 with a common density p(x) on Rn. We study the moments of the nearest neighbor distance. The quantity (1) appears commonly in the literature on random geometric graphs, where directed and undirected nearest neighbor graphs are analyzed as special cases of more general frameworks [9, 10, 13]. The nearest neighbor distance serves as the quantity of interest with the hope that in the future, the ideas can be represented in a more abstract form. Nearest neighbor distances and distributions play a major role in the understanding of nonparametric estimation in general [1, 4]. 1 y dy, where the definition of g depends on n, k and α (see Section 3)
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